Microfluidic waves

The propagation of pressure waves in fluidic channels with elastic covers is discussed in view of applications to flow control in microfluidic devices. A theory is presented which describes pressure waves in the fluid that are coupled to bending waves in the elastic cover. At low frequencies, the lateral bending of the cover dominates over longitudinal bending, leading to propagating, non-dispersive longitudinal pressure waves in the channel. The theory addresses effects due to both the finite viscosity and compressibility of the fluid. The coupled waves propagate without dispersion, as long as the wave length is larger than the channel width. It is shown that in channels of typical microfluidic dimensions, wave velocities in the range of a few 10 m s(-1) result if the channels are covered by films of a compliant material such as PDMS. The application of this principle to design microfluidic band pass filters based on standing waves is discussed. Characteristic frequencies in the range of a few kHz are readily achieved with quality factors above 30.     

Thermocapillary convection in double-layer fluid structures within a two-dimensional open cavity

Thermocapillary convection within a differentially-heated open rectangular cavity containing two immiscible liquid layers is considered in the absence of gravitational effects. The temperature and flow fields in the two layers are computed using domain mapping in conjunction with a finite-difference scheme on a staggered grid. The melt-encapsulant and air-encapsulant interfaces are allowed to deform, with the contact lines pinned on the solid boundaries. The presence of a free surface at the top leads to increased convection in the encapsulant phase while retarding thermocapillary flow in the melt. The intensity of thermocapillary convection in the encapsulated layer is reduced as the viscosity of the encapsulant is increased or the thickness of the encapsulant layer is decreased. Choosing an encapsulant with a greater sensitivity of interfacial tension to temperature (as compared to that of the melt phase) can almost completely suppress thermocapillary convection in the melt. Deformations of the melt-encapsulant interface in an open cavity are found to be larger than those in a closed cavity with a rigid top surface, due to higher pressure gradients realized in the encapsulant phase. In contrast to interface deformation behavior reported earlier for a double-layer system in a closed cavity, the shape of the melt-encapsulant interface is qualitatively similar for all values of the viscosity ratio, with the interface dipping into the melt near the cold wall, and into the encapsulant near the hot wall. For the double-layers considered in this study, a free surface at the top of the encapsulant layer was found to be more effective than a rigid top in reducing the intensity of thermocapillary convection in the melt.     

On the Mach- and Reynolds-Number Dependence of the Flat-Plate Turbulent Boundary Layer Wall-Pressure Spectrum

Wall-pressure measurements are performed using an array of pinhole microphones mounted into the test wall of a low noise, low turbulence flow facility. The frequency–wave-number spectra for both streamwise and spanwise directions are obtained through the spatial Fourier transformation of cross-spectral measurements in the said directions. The results of the wind-tunnel measurements reveal an interesting behavior of the low wave-number portion of the wall-pressure spectrum. Namely, this portion of the spectrum does not exhibit any Mach-number dependence for nearly incompressible flows. A low Mach-number analysis accounting for finite Reynolds-number effects is performed. This analysis shows the wall-pressure spectrum to have a leading term independent of the Mach number. The frequency dependence of the theoretical spectrum agrees well with that of the experimental spectrum. The physical mechanism for the Mach-number independence of the spectrum is attributed to a hydrodynamic dipole contribution to the wall pressure resulting from the Stokes layer induced by the turbulent eddies. The higher differential order of the viscous equations of motion allows for the existence of such Stokes layers. They have no counterpart in invisicd theories wherein no such Mach-number independence of the spectrum has been observed. A simple model problem is devised which isolates the viscous mechanism responsible for the modification of the wall pressure and the corresponding sound field. It is argued that the physical process involved is akin (albeit inversely) to that operative in the boundary-layer receptivity problem for nearly incompressible flows. Received 1 June 1998 and accepted 25 November 1998     

Effect of free surface heat transfer on thermocapillary flow in double-layer fluid structures

In this paper, we examine the effect of heat transfer at the free encapsulant-air surface on thermocapillary flow in a rectangular melt-encapsulant double-layer fluid structure. We show that increased heat transfer to the double-layer system through the free surface weakens thermocapillary convection in the encapsulant phase and enhances the convection in the melt phase.     

Stability of the Shape of a Viscous Drop under Buoyancy-Driven Translation in a Hele-Shaw Cell

It has been shown (N. R. Gupta, A. Nadim, H. Haj-Hariri, and A. Borhan, J. Colloid Interface Sci. 218, 338 1999) that a circular drop translating in a Hele-Shaw cell under the action of gravity is linearly stable for nonzero interfacial tension. In this paper, we use the boundary integral method to examine the nonlinear evolution of the shape of initially noncircular drops translating in a Hele-Shaw cell. For prolate initial deformations, it is found that the drop reverts to a circular shape for all finite Bond numbers considered. Initially oblate drops, on the other hand, are found to become unstable and break up if the initial shape perturbation is of sufficiently large magnitude. The critical conditions for the onset of drop breakup are examined in terms of the magnitude of the initial deformation as a function of Bond number. Two branches of marginal stability are identified and the effects of viscosity ratio and asymmetric initial perturbations on the stability diagram are discussed. Copyright 2000 Academic Press.     

A numerical study of the effect of insoluble surfactants on the stability of a viscous drop translating in a Hele-Shaw cell

A circular drop is a linearly stable solution for the buoyancy-driven motion of drops in a Hele-Shaw cell [Gupta et al. J. Colloid Interface Sci.218(1), 338 (1999)]. In the absence of surface-active agents, an initially prolate drop always goes to a steady circular shape while initially oblate drops exhibit complex dynamics [Gupta et al. J. Colloid Interface Sci.222, 107 (2000)]. In this study, the effect of insoluble surfactant impurities on the critical conditions for drop breakup is explored by using the Langmuir adsorption framework in conjunction with a physically based expression for the depth-averaged tangential stress exerted on a two-phase interface in a Hele-Shaw cell. It is shown that the presence of surfactants can have both a stabilizing and a destabilizing effect on the shape of the drop, depending on the Bond number, the magnitude of the initial perturbation, and the strength of surface convection. Similar to the clean drop dynamics, two marginally stable branches are found. Increasing the surface Peclet number results in the stabilization of the main branch while the secondary branch shifts to higher Bond numbers. The mode of breakup is also found to be strongly influenced by the strength of surface convection.     

Effect of interface deformability on thermocapillary motion of a drop in a tube

The effect of an externally imposed axial temperature gradient on the mobility and deformation of a drop in an otherwise stagnant liquid within an insulated cylindrical tube is investigated. In the absence of bulk transport of momentum and energy, the boundary integral technique is used to obtain the flow and temperature fields inside and outside the deformable drop. The steady drop shapes and the corresponding migration velocities are examined over a wide range of the dimensionless parameters. The steady drop shape is nearly spherical for dimensionless drop sizes <0.5, but becomes slightly elongated in the axial direction for drop sizes comparable to tube diameter. The adverse effect of drop deformation on the effective temperature gradient driving the motion is slightly more pronounced than its favorable effect of reducing drag, thereby leading to a slight reduction in drop mobility with increasing drop deformation. Increasing the viscosity ratio reduces drop deformation and leads to a slight enhancement in the relative mobility (with respect to free thermocapillary motion) of confined drops. When the drop fluid has a lower thermal conductivity than the exterior phase, the presence of the thermally-insulating wall increases the thermal driving force for drop motion (compared to that for the same drop in unbounded domain) by causing more pronounced bending of the isotherms toward the drop. However, the favorable thermal effect of the confining wall is overwhelmed by its retarding hydrodynamic effect, causing the confined drop to always move slower than its unbounded counterpart regardless of the value of the thermal conductivity ratio.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

Characteristics Analysis of the Parabolized Stability Equations

Some mathematical details of the parabolized stability equations (PSE) [4] are investigated. In particular, the sources of (unwanted) ellipticity in these equations are identified and suggestions are made for their suppression. Both the compressible and incompressible equations in primitive-variable formulation are discussed. Remarks are also made on the velocity-vorticity formulation. A slight modification to the PSEs usually method of treatment of streamwise derivatives higher than first is proposed. Also, an expression is derived for limiting the streamwise gradient of the pressure shape function.     

An Embedding Method for Bluff Body Flows: Interactions of Two Side-by-Side Cylinder Wakes

We introduce a simple method for the numerical simulation of bluff body flows where the solid object is represented by a distributed body force in the Navier–Stokes equations. The body force density is found at every time step to reduce the velocity within the computational cells occupied by the rigid body to a prescribed value. The method combines certain ideas from the immersed boundary method which was developed to treat biofluid mechanical flows and the volume-of-fluid method for simulating flows with fluid–fluid interfaces. The main advantage of this embedding method is that the computations can be effected on a regular Cartesian grid, without the need to fit the grid to the bluff body surfaces. Thus, flow past several complex bodies can be treated as easily as flow past a single body. The method is validated by reproducing well-established results for vortex shedding from a stationary cylinder. The flow past two side-by-side cylinders is then investigated. When the distance between the cylinders is small, they are seen to shed vortices in-phase, whereas for larger distances, the shedding occurs in anti-phase. For intermediate distances, various shedding patterns are observed, including quasi-periodic, asymmetric and chaotic regimes. Mean values and phase portraits associated with the cylinder lift and drag coefficients, as well as spectral analysis of the same data, are used to describe the flow. A transition diagram that can be compared with experiments or models outlines the various dynamical regimes as a function of the distance between the cylinders and the Reynolds number.